Following are final exam scores, arranged in increasing order, for students in an introductory statistics course.

58
60
61
63
66
68
69
70
71
71
72
75
76
76
76
77
77
78
83
84
84
87
87
89
91
95
96
96
97
98

(a)Find the first quartile of the scores.
The first quartile of the scores is

To find the first quartile (Q1) of the scores, we follow these steps:

### Step 1: Arrange the data in ascending order.
This has already been done for us:

\[ 58, 60, 61, 63, 66, 68, 69, 70, 71, 71, 72, 75, 76, 76, 76, 77, 77, 78, 83, 84, 84, 87, 87, 89, 91, 95, 96, 96, 97, 98 \]

### Step 2: Determine the number of data points (\(n\)).
There are \(n = 30\) data points.

### Step 3: Find the position of Q1.
The first quartile (Q1) is the value at the 25th percentile. The position of Q1 can be found using the formula:

\[
Q1 = \frac{n + 1}{4}
\]

Substituting the value of \(n\):

\[
Q1 = \frac{30 + 1}{4} = \frac{31}{4} = 7.75
\]

### Step 4: Locate Q1.
Since \(Q1\) is at position 7.75, it lies between the 7th and 8th values in the data set. To find the exact value, we can interpolate between the 7th and 8th values.

- The 7th value is 69.
- The 8th value is 70.

The value of Q1 can be found by interpolating between these two values:

\[
Q1 = 69 + 0.75 \times (70 - 69) = 69 + 0.75 = 69.75
\]

### Final Answer:
The first quartile (Q1) of the scores is **69.75**.

Would you like more details on the interpolation process or have any other questions?

Here are five related questions:
1. How do you find the median of a dataset?
2. What is the difference between the first quartile and the median?
3. How do you calculate the third quartile (Q3) of the same dataset?
4. How is the interquartile range (IQR) determined?
5. What are the implications of a large IQR in a dataset?

**Tip:** Quartiles help in understanding the distribution of data, particularly when identifying the spread and detecting outliers.

Discover how to find the first quartile (Q1) of exam scores in an introductory statistics course. Step-by-step solution provided with detailed interpolation method. Suitable for high school students studying statistics.

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